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Badosa and Kontaveit both got as high as number two in the world rankings last year, but neither currently sits in the top 10, with Badosa down at number 11 and Kontaveit 17th.
Tuesday's tussle in South Australia went the way of the Spaniard, with Badosa scoring a 6-4 6-3 win against her Estonian opponent.
She had five aces and only one double fault, winning 77 per cent of points when landing a first serve in.
Another Estonian awaits her at the last-16 stage in veteran campaigner Kaia Kanepi and Badosa is happy with how she had begun the tournament.
"I think I started really good," Badosa said in an on-court interview. "I was serving very well. I was going for the first shots."
Asked about signs of improvement with her serve, Badosa said: "It depends on the day, to be honest. I worked a lot this past 10 days and was really focused on that, so I'm happy about that."
Czech Barbora Krejcikova, another player with a career-high ranking of number two, earned a second-round shot at Russian Daria Kasatkina after winning 6-2 7-6 (7-3) against last year's runner-up Alison Riske-Amritraj.
Other winners in Adelaide on Tuesday were Amanda Anisimova, Zheng Qinwen, Veronika Kudermetova, Anna Kalinskaya and Katerina Siniakova.
At the Hobart International, second seed Elise Mertens fell 6-4 6-4 to fellow Belgian Maryna Zanevska, while French third seed Alize Cornet was also knocked out, losing 6-4 6-2 to Italy's Elisabetta Cocciaretto.
Former Australian Open champion Sofia Kenin had no such trouble, seeing off China's Zhu Lin 6-2 6-2